Arithmetic and geometric sequences: where does their name come from?

Question

Where does the name of these two famous types of sequences come from?

The article Geometric progression of Wikipedia says that the geometric sequence is called like this because every term is the geometric mean of its two adjacent terms. Though it is true, it only reduces the question to: why is the geometric mean geometric (in opposition to the arithmetic mean).

Continuing my investigation on Geometric mean, I was told that a square with the same area than a rectangle with sides $a$ and $b$ has their geometric mean $\sqrt{ab}$ for side. That's again totally true, but a square with same perimeter than this rectangle of sides $a$ and $b$ has their arithmetic mean $\frac{a+b}{2}$ for side!

Thus, my question: Who coined these names? And why? Why is the geometric mean more geometric than the arithmetic mean?

Answer 1

Who coined these names? And why?

Who is already very difficult to know. Why is nearly impossible. Because there was no name for the mean?

Anyway, according to a book by Anthony Lo Bello (1), "arithmetic" comes from the Greek word ἀριθμός arithmos, meaning "number". In a similar way, "geometric" comes from γεωμετρία geometría, "measurement of the earth". [The transliterations are not from the book.]

This section, written by Amartya Dutta of the Indian Statistical Institute, mentions the term "arithmetic mean" was used in 1635 by Henry Gellibrand, an astronomer. However, I didn't find it. The Gaugers Magazine, written by William Hunt and published in 1687, contains the earliest use I could verify.

"Geometric mean" was already used in the 1771 edition of the Encyclopædia Britannica, by James A. Landau (see here). The actual term ("geometrical mean") comes from a long-titled work written by E. Halley and published from 1695 to 1697, in a volume of the magazine Philosophical Transactions. You can see the unformatted original text here.

In another question, David K shows a wonderful figure illustrating why this mean is so geometric.